= Solution
Assume the riskless gross return $q=1+r>0$, as required by the positive-wealth model, and a finite moment $m_b=\mathbb EX_0^b$. Define
$$
C(\theta)=\mathbb E[\theta X_0+(1-\theta)q]^b,\qquad
C_* =\max_{\alpha\leq\theta\leq1}C(\theta).
$$
This maximum exists: the control interval is compact and dominated convergence gives continuity, using the integrable bound $(X_0+q)^b\leq X_0^b+q^b$ for $0<b\leq1$.
The terminal value is $V_N(w)=U(w)$. If $V_{n+1}(w)=a_{n+1}U(w)$, the <Bellman equation> and <independence> of the next return give
$$
V_n(w)=\max_{\theta\in[\alpha,1]}\mathbb E V_{n+1}(w[\theta X_n+(1-\theta)q])
=a_{n+1}U(w)C_*.
$$
Backward induction therefore proves
$$
\boxed{V_n(w)=a_nU(w),\qquad a_n=C_*^{N-n}.}
$$
Any maximizing proportion can be used each day; allowing adapted choices cannot exceed the Bellman bound. For $\alpha=1$, $C_*=m_b$, exactly agreeing with part (b). If $m_b=\infty$ and at least one day remains, the allowed choice $\theta=1$ already gives infinite expected utility, so the value is infinite rather than a finite coefficient. This is the necessary moment qualification to the finite optimization formula.
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